Method and apparatus for wavefront sensing

ABSTRACT

A method of measuring characteristics of a wavefront of an incident beam includes obtaining an interferogram associated with the incident beam passing through a transmission mask and Fourier transforming the interferogram to provide a frequency domain interferogram. The method also includes selecting a subset of harmonics from the frequency domain interferogram, individually inverse Fourier transforming each of the subset of harmonics to provide a set of spatial domain harmonics, and extracting a phase profile from each of the set of spatial domain harmonics. The method further includes removing phase discontinuities in the phase profile, rotating the phase profile, and reconstructing a phase front of the wavefront of the incident beam.

CROSS-REFERENCES TO RELATED APPLICATIONS

This application claims priority to U.S. Provisional Patent ApplicationNo. 61/923,362, filed on Jan. 3, 2014, entitled “Apparatus and Methodfor Wavefront Sensing,” the disclosure of which is hereby incorporatedby reference in its entirety for all purposes.

BACKGROUND OF THE INVENTION

Numerous techniques have been used for wavefront sensing. Generally,wavefront sensors are used to measure aberrations in an opticalwavefront.

Although various techniques have been used to implement wavefrontsensors that measure the characteristics of wavefronts, there is a needin the art for improved methods and systems related to wavefrontsensing.

SUMMARY OF THE INVENTION

Embodiments of the invention generally relate to the field of opticalwavefront sensing. More particularly, the methods and apparatusdescribed herein relate to enhanced wavefront characterization using,for example, a checkerboard amplitude mask based on the principle oflateral shearing interferometry (LSI). In a particular embodiment,methods and apparatus are provided that overcome the spatial resolutionlimit associated with conventional wavefront sensing techniques, forexample, lateral shearing interferometry (LSI).

According to an embodiment of the present invention, a method ofmeasuring characteristics of a wavefront of an incident beam isprovided. The method includes obtaining an interferogram associated withthe incident beam passing through a transmission mask and Fouriertransforming the interferogram to provide a frequency domaininterferogram. The method also includes selecting a subset of harmonicsfrom the frequency domain interferogram, individually inverse Fouriertransforming each of the subset of harmonics to provide a set of spatialdomain harmonics, and extracting a phase profile from each of the set ofspatial domain harmonics. The method further includes removing phasediscontinuities in the phase profile, rotating the phase profile, andreconstructing a phase front of the wavefront of the incident beam.

According to another embodiment of the present invention, a method forperforming optical wavefront sensing. The method includes providing anamplitude transmission mask having a light input side, a light outputside, and an optical transmission axis passing from the light input sideto the light output side. The amplitude transmission mask ischaracterized by a checkerboard pattern having a square unit cell ofsize Λ. The method also includes directing an incident light fieldhaving a wavelength λ to be incident on the light input side andpropagating the incident light field through the amplitude transmissionmask. The method further includes producing a plurality of diffractedlight fields on the light output side and detecting, at a detectordisposed a distance L from the amplitude transmission mask, aninterferogram associated with the plurality of diffracted light fields.The distance satisfies

${0 < L < {\frac{1}{8}\frac{\Lambda^{2}}{8}\mspace{14mu} {or}\mspace{14mu} \frac{1}{4}\frac{\Lambda^{2}}{\lambda}( {{2n} - 1} )} < L < {\frac{1}{4}\frac{\Lambda^{2}}{\lambda}( {{2n} + 1} )}},$

where n is an integer greater than zero.

According to a specific embodiment of the present invention, a wavefrontsensor is provided. The wavefront sensor includes an amplitude-onlytransmission mask characterized by a checkerboard pattern having a unitsquare cell of size Λ and a detector disposed at a distance, L,optically downstream of the amplitude-only transmission mask. Thedistance satisfies

${0 < L < {\frac{1}{8}\frac{\Lambda^{2}}{8}\mspace{14mu} {or}\mspace{14mu} \frac{1}{4}\frac{\Lambda^{2}}{\lambda}( {{2n} - 1} )} < L < {\frac{1}{4}\frac{\Lambda^{2}}{\lambda}( {{2n} + 1} )}},$

where λ is the wavelength of the incident field and n is a positiveinteger. The wavefront sensor also includes a computer coupled to thedetector.

According to an embodiment of the present invention, a compact wavefrontmeasurement system and data analysis method is provided. The wavefrontsensor includes a checkerboard amplitude-only transmission mask disposeda predetermined distance in front of a detector. An incident wavefront,which is defined by a phase front and an intensity profile, isdiffracted through the checkerboard mask. An image of self-interferencebetween the replicated diffracted beams is detected. The frequencydomain analysis of the diffraction pattern created by the checkerboardmask enables the extraction of a high resolution wavefront map in theincident field. This extraction process involves a Fourier domainstructure of the cross-terms between the diagonal and zero-orderdiffraction fields belonging to the checkerboard mask.

Numerous benefits are achieved by way of the present invention overconventional techniques. For example, embodiments of the presentinvention provide methods and systems that provide physically andmathematically simpler apparatus in comparison to conventional systems.Moreover, embodiments of the present invention provide improvements inmeasurement results, for example, higher spatial resolution, easiermanufacturability, and size scalability, than conventional systems.These and other embodiments of the invention along with many of itsadvantages and features are described in more detail in conjunction withthe text below and attached figures.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a perspective diagram illustrating elements of a wavefrontsensing system according to an embodiment of the present invention.

FIG. 2 is a side-view of elements of a wavefront sensing systemaccording to an embodiment of the present invention.

FIG. 3A is diagram illustrating a Hartmann mask.

FIG. 3B is a frequency domain plot of the intensity associated with theHartmann mask illustrated in FIG. 3A.

FIG. 4A is a diagram illustrating checkerboard pattern transmissionmask.

FIG. 4B is a frequency domain amplitude plot of the amplitude associatedwith the Checkerboard pattern transmission mask illustrated in FIG. 4A.

FIG. 4C is a frequency domain plot of intensity associated with theCheckerboard pattern transmission mask illustrated in FIG. 4A.

FIG. 5A is a diagram illustrating a unit cell of the checkerboardpattern transmission mask illustrated in FIG. 4A.

FIG. 5B is a frequency domain plot of associated with a measuredinterferogram associated with the rotated checkerboard patterntransmission mask illustrated in FIG. 4A.

FIG. 6A is a plot illustrating a wavefront measurement made using anembodiment of the present invention.

FIG. 6B is a plot illustrating a wavefront measurement made using aconventional Shack-Hartmann sensor.

FIG. 6C is a plot illustrating the difference between the plots in FIGS.6A and 6B.

FIG. 7 is a diagram illustrating the use of an embodiment of the presentinvention in aligning hexagonally segmented mirrors according to anembodiment of the present invention.

FIG. 8 is a method of measuring a wavefront according to an embodimentof the present invention.

FIG. 9 is a schematic diagram illustrating a wavefront sensor accordingto an embodiment of the present invention.

FIG. 10A is a plot illustrating a phase profile for a first harmonicaccording to an embodiment of the present invention.

FIG. 10B is a plot illustrating an unwrapped phase profile for the firstharmonic according to an embodiment of the present invention.

FIG. 10C is a plot illustrating a rotated phase profile for the firstharmonic according to an embodiment of the present invention.

FIG. 11A is a plot illustrating a phase profile for a second harmonicaccording to an embodiment of the present invention.

FIG. 11B is a plot illustrating an unwrapped phase profile for thesecond harmonic according to an embodiment of the present invention.

FIG. 11C is a plot illustrating a rotated phase profile for the secondharmonic according to an embodiment of the present invention.

DETAILED DESCRIPTION OF SPECIFIC EMBODIMENTS

Various techniques have been used to perform wavefront sensing.Heterodyne interferometers such as Michelson, Fizeau, or Mach-Zehnderschemes can require separate reference beams and are bulky. Homodyneinterferometers based on interference between the original field and itsspatially sheared replicas are called lateral shearing interferometers.Hartmann sensors, Ronchi tests, Shack-Hartmann sensors, like lateralshearing interferometers, do not require on-line reference beams.Hartmann sensors use a two-dimensional array of holes whose dimensionsare well known. The projection of the image of the holes on a detectorfurther away carries the spatial derivative information from which thewavefront can be reconstructed. Shack-Hartmann sensors improve the lowsignal-to-noise ratio in the Hartmann sensor by using an array ofmicrolens instead of holes. The deviation of the centroid of each focalspot from the micro-lens, as in Hartmann sensors, contains spatialslopes information that can be integrated to reconstruct a wavefrontmap.

Shack-Hartmann wavefront sensors are used in diverse fields, includingastronomy, eye diagnostics, and laser beam correction. However, thesewavefront sensors have the disadvantage of not being able to measurerelative piston terms coming from segmented optics. Incoherentbackground noise is another problem with this type of sensor sincebackground room light distorts the centroiding of each microlensletcell.

An alternative to this type of wavefront sensing is to use the techniqueof lateral shearing interferometry (LSI), which is not affected byincoherent background light and can be used to detect relative pistonterms. One version of a lateral shearing interferometer uses a specialtwo-dimensional amplitude-phase grating (such as a Hartmann orShack-Hartmann mask as illustrated in FIG. 3A) and is based onmulti-wave self-interference coming from the grating. This approachrequires using both a phase and amplitude grating, which is relativelyexpensive to manufacture and not easily scalable to a larger size.

The transmission function of a periodic mask (grid) where Λ_(x), Λ_(y)are periods in the x and y directions can be written as

$\begin{matrix}{{{T( {x,y} )} = {\sum\limits_{m,n}{c_{m,n}{{\exp ( {{\; \frac{2\pi}{\Lambda_{x}}{mx}} + {\; \frac{2\pi}{\Lambda_{x}}{ny}}} )}.}}}},} & (1)\end{matrix}$

where c_(m,n) are the Fourier coefficients that uniquely represent aparticular two dimensional periodic structure such as a grid or theHartmann mask as illustrated in FIG. 3A. In general, the Fouriercoefficients (c_(m,n)) are complex numbers. The c_(m,n) of a Hartmannmask, for example, are distributed at all indices of (m,n) as shown inFIG. 3B. Each grid in FIG. 3B is in units of inverse period of the(Shack-) Hartmann cell.

Each term in the summation in Eq. (1) with m≠0 or n≠0 can be referred toas a harmonic or a carrier and the term with m=0 and n=0 as a DC term.As any one-dimensional periodic structure can be represented by the sumof a DC term and harmonics, the two-dimensional periodic structure suchas the Hartmann mask can also be represented by the sum of a DC term andtwo-dimensional harmonics that has two fundamental frequencies in the x-and y-directions, respectively.

Electric-field propagation over a distance L through a grid or mask canbe similarly represented as a Fourier series. When the electromagneticfield is detected on a CCD after propagating through a Hartmann mask,for example, its Fourier coefficients are multiplied by propagationfactors (A_(m,n)) according to the propagation distance L and thewavelength, and the time integrated intensity (not the E field) isdetected at the detection plane. The mathematical expression on thedetector follows a form of the absolute squared of the above expressionwith c_(mn) replaced with c_(mn)A_(mn):

$\begin{matrix}{{{H( {x,y} )} = {\sum_{m,n,m^{\prime},n^{\prime}}{c_{m,n}c_{m,n}^{*}A_{m,n}A_{m,n}^{*}{\exp \lbrack {\frac{2\pi \; {( {m - m^{\prime}} )}x}{\Lambda_{x}} + \frac{2\pi \; {( {n - n^{\prime}} )}y}{\Lambda_{y}}} \rbrack}}}},,} & (2)\end{matrix}$

where the asterisk denotes a complex conjugation operation. Thedifference indices (m−m′) and (n−n′) can be graphically understood asthe distance between the dots in FIG. 3B. As described more fully below,the Fourier representation of the intensity of the E field, i.e., themodulus squared of the field for a Hartmann mask, presents a complicatedstructure. Even the first order term corresponding (m−m′, n−n′)=(1,0)has an infinite sum of other harmonics. One of these sets can be {(m=1,n=0), (m′=0, n′=0)} or {(m=2, n=−3), (m′=1, n′=−3)}, etc.

As described herein, embodiments of the present invention extract thewavefront based on orthogonal harmonics in the frequency domain, i.e.,the open or white dots 407 a, 407 b, 407 c, and 407 d in FIG. 4C nearestthe origin, for example, located at an angle (e.g., 45 degrees) from thex and y axes. The use of a polynomial fit to the recovered wavefrontslopes only provides a relatively low resolution wavefront map. In someconventional methods, the harmonics used in the Fourier analysis methodsis only a first order approximation. This approximation prevents furtherimprovement in the spatial resolution of the wavefront beingreconstructed.

FIG. 1 is a perspective diagram illustrating elements of a wavefrontsensing system according to an embodiment of the present invention.Referring to FIG. 1, the wavefront sensor 100 provided by embodiments ofthe present invention includes a number of elements. Incident wavefront101, with a generally non-planar phase front, is represented by wavylines as appropriate to a non-planar wavefront. The wavefront 101 passesthrough and is diffracted by amplitude-only checkerboard patterntransmission mask 102 into a zero-order (arrow 104) and four diagonalfirst-order terms (arrows 105 a, 105 b, 106 a, and 106 b) in theembodiment illustrated in FIG. 1, which can be considered as four edgesof a pyramid. The first-order terms, also referred to as first-orderdiffracted orders, are illustrated in FIG. 1 by two arrows lying on thediagonal edges (arrows 105 a and 105 b) and two arrows lying on theanti-diagonal edges (arrows 106 a and 106 b). Thus, two dimensionaldiffraction is illustrated in FIG. 1.

In some implementations, the checkerboard pattern mask is anamplitude-only mask, fabricated, for example, by patterning a chromelayer deposited on a fused silica or other suitable substrate. In theseimplementations, the phase impact of the light passing through thesubstrate on which the pattern is formed is negligible. The higher-orderdiffraction terms are not shown in FIG. 1 for purposes of clarity. Thediffracted zero-order beam 104 and the diffracted first-order beams 105a/105 b and 106 a/106 b interfere together at a predetermined distancefrom the mask 102 and form interferogram 108 at a detector plane 110 ofan imaging device 112, which can correspond to the imaging plane of adetector, such as a charge-coupled-device (CCD) camera.

FIG. 2 is a side-view of elements of a wavefront sensing systemaccording to an embodiment of the present invention. The wavefrontsensing system illustrated in FIG. 2 corresponds to the wavefrontsensing system illustrated in FIG. 1, but presented in one dimension. Itwill be appreciated that an analogous diagram could be prepared the forthe orthogonal direction. As discussed above, for purposes of clarity inFIGS. 1 and 2, the optical operation of the wavefront sensor isillustrated only considering the zero-order and first-order diffractionterms (two diffracted orders in each of the x and y directions). Thediffraction of the wavefront 101 through the checkerboard mask 102 canbe considered (in one dimension) as propagation of the zero-orderdiffracted beam 204 and two first-order (205 a and 205 b) diffractedbeams. The zero-order beam 204 propagates in the same direction as theincident beam. These diffracted beams (204/205 a/205 b) correspond tothe diffracted orders (104/105 a/105 b) discussed in relation to FIG. 1.

The first-order beams 205 a and 205 b propagate at an angle with respectto the zero-order term. Immediately after the amplitude-onlycheckerboard mask 102, the three fields are overlapped. At a distance Laway, they separate from each other because they are propagating indifferent directions. These laterally separated beams propagating at anangle with respect to each other, interfere together to form aninterferogram 108 at the detector plane 110 of the imaging device 112,at which a CCD or other suitable detector can be placed. The 1-D pictureof the interferogram illustrated in FIG. 2 can be analogouslyrepresented by an FM radio signal in two dimensions in which themodulation frequency is varied locally. In other words, the interferencepattern is a result of the local frequency that is modulated as afunction of the features of the incoming wavefront. As described herein,embodiments of the present invention analyze the local frequency shiftin two dimensions to extract the phase information associated with theincoming wavefront 101.

FIG. 3A is diagram illustrating a Hartmann mask featuringsquare/rectangular apertures arrayed on a dark background. Roundapertures can be used as well. In some implementations, this mask isreferred to as a Shack-Hartmann mask. FIG. 3B is a frequency domain plotof the intensity associated with the Hartmann mask 302 illustrated inFIG. 3A. As illustrated in FIG. 3B, the distribution of diffracted fieldor harmonics 301 in the frequency domain is periodic in the spatialfrequency domain as appropriate to the Hartmann (or Shack-Hartmann) maskillustrated in FIG. 3A. A Fourier transform of the Hartmann maskillustrated in FIG. 3A produces the harmonics existing at thefrequencies illustrated in FIG. 3B.

FIG. 4A is a diagram illustrating checkerboard pattern transmissionmask. As illustrated in FIG. 4A, an amplitude-only transmission mask isutilized as an element of the wavefront sensor in some embodiments.

FIG. 4B is a frequency domain amplitude plot (i.e., related to theelectric field) associated with the Checkerboard pattern transmissionmask illustrated in FIG. 4A. In the frequency domain, the Fouriertransform of the amplitude only transmission mask produces amplitudepeaks 404 at frequency values associated with the periodicity of themask. In particular, the harmonics 405 a and 405 b are the harmonicsclosest to the origin 402.

As illustrated in FIG. 4B in comparison with FIG. 3B, the checkerboardpattern mask has fewer spatial frequency components in the spatialfrequency domain than those associated with the Hartmann mask.Accordingly, as illustrated in FIG. 4B, in the frequency domain, anumber of the multiples of the fundamental frequency components aremissing in the frequency domain, providing a sparser frequency domainplot.

FIG. 4C is a frequency domain plot of intensity associated with theCheckerboard pattern transmission mask illustrated in FIG. 4A. Becausethe intensity is proportional to the square of the amplitude (e.g. themodulus squared), the intensity distribution is periodic, similar to thefrequency domain amplitude plot in FIG. 4B. In FIG. 4C, the harmonics at407 a, 407 b, 407 c, and 407 d are highlighted by an open circle. Thedifference in shading in this figure is merely used to represent thatharmonics 407 a-407 d are of particular interest for analysis. Inparticular, to generate the harmonic 407 a, only three values areneeded, greatly simplifying the analysis: the origin 402, the harmonic405 a in the first quadrant closest to the origin, and the harmonic 405c in the third quadrant closest to the origin. Likewise, to generate theharmonic 407 b, only three values are needed: the origin 402, theharmonic 405 b in the first quadrant closest to the origin, and theharmonic 405 d in the third quadrant closest to the origin.

As discussed above, T(x, y) is the transmission function of a periodicmask and Λ_(x), Λ_(y) are periods in x and y directions. A twodimensional periodic structure can be uniquely represented by itsFourier coefficients c_(m,n) as expressed in Eq. (1). Contrary to thedense distribution of the Fourier coefficients (301, as shown in FIG.3B) for a conventional Hartmann mask (302 as shown in FIG. 3A) or for aconventional Shack-Hartmann lenslet array, the Fourier coefficients,c_(m,n) 404, belonging to an amplitude-only checkerboard pattern 102 asshown in FIG. 4A, are all zeros at even indices except at (0,0).Non-zero terms are shown as frequency domain amplitude peaks 404 in FIG.4B.

With further reference to Eq. (2) above, the difference indices (m−m′)and (n−n′) can be graphically understood as the distance between thepeaks 404 in FIG. 4B and dots 301 in FIG. 3B. For an amplitude-onlycheckerboard mask 102, the difference indices have non-zero terms onlyat (m−m′, n−n′)={(0,0), (1,1), (1,−1), (−1,1), (−1,−1), (2,0), (2,2),(0,2), (−2,2), (−2,0), (−2,−2), (0,−2), . . . }. The locations of thenon-zero terms are shown in FIG. 4C. The (m−m′, n−n′)=(0,0) term comesfrom multitudes of sets such as {(m=0, n=0), (m′=0, n′=0)} or {(m=1,n=−2), (m′=1, n′=−2)}. There are an infinite number of sets contributingto the (m-m′, n-n′)=(0,0) term, but in practice the actual number ofterms are limited due to physical optics.

On the other hand, the term corresponding to (m−m′, n−n′)=(1,1) comesfrom only two terms, that is, {(m=1, n=1), (m′=0, n′=0)} and {(m=0,n=0), (m′=−1, n′=−1)}. This corresponds to the peak 407 a in the firstquadrant of FIG. 4C. Likewise, the other peaks 407 b, 407, and 407 dcorrespond to (m−m′, n−n′)=(−1,1) or (−1,−1) or (1,−1) and all theseterms involve the sum of only two terms each. The other higher-orderterms involve more than two terms. The Fourier representation of theintensity of the field, that is, the modulus squared of the field for aHartmann mask has no such simple structure. Even the first-order termcorresponding (m−m′, n−n′)=(1,0) has an infinite sum of other harmonics.One of these sets can be {(m=1, n=0), (m′=0, n′=0)} or {(m=2, n=−3),(m′=1, n′=−3)}, etc. The simplicity in the number of fields interactingin the diagonal terms of intensity measured through the amplitude-onlycheckerboard mask (peaks 407 a-407 d in FIG. 4C provides the benefitsand advantages enabled by the present invention.

The Fourier transform of the measured interferogram has peaks inlocations as shown in FIG. 4C. As described more fully below,embodiments of the present invention utilize the diagonal terms (peaks407 a-407 d in FIG. 4C). The intensity (i.e., the magnitude squared) ofthe incident field is

I=¼(|H _(1,1) |+|H _(−1,1) +|H _(1,−1) |+|H _(−1,−1)|)  (3)

The phase of these terms after inverse Fourier transformation have phasedifference information of the incident wavefront in x- and y-directions.They can be expressed as

$\begin{matrix}{{{{\frac{\partial}{\partial x}\phi} = {{- \frac{\Lambda_{x}}{4\lambda \; L}}( {{\arg \; H_{1,1}} - {\arg \; H_{{- 1},1}} + {\arg \; H_{1,{- 1}}} - {\arg \; H_{{- 1},{- 1}}}} )}}{\frac{\partial}{\partial y}\phi} = {{- \frac{\Lambda_{y}}{4\lambda \; L}}( {{\arg \; H_{1,1}} + {\arg \; H_{{- 1},1}} - {\arg \; H_{1,{- 1}}} - {\arg \; H_{{- 1},{- 1}}}} )}},,} & (4)\end{matrix}$

where φ is the incident phase front in units of radians and ‘arg’denotes the operation of taking phase of a complex number. H(±1, ±1) arethe inverse Fourier transformed diagonal terms as shown by peaks 407a-407 d FIG. 4C. The phase can be reconstructed by integrating thederivatives in Eq. (4) as described more fully below. The processdescribed so far cannot resolve the wavefront map beyond the spatialresolution limit of shear (λL/Λ) imposed by conventional lateralshearing interferometry schemes.

Some embodiments of the present invention improve the wavefrontresolution by point-by-point optimization as follows:

As described in regard to FIGS. 4A-4C, each diagonal term of theamplitude-only checkerboard mask interferogram is made of the sum ofonly two terms:

H(1,1)˜E(1,1)E(0,0)*+E(0,0)E(−1,−1)*

H(−1,1)˜E(−1,1)E(0,0)*+E(0,0)E(1,−1)*

H(−1,−1)˜E(−1,−1)E(0,0)*+E(0,0)E(1,1)*

H(1,−1)˜E(1,−1)E(0,0)*+E(0,0)E(−1,1)*.  (5)

where E(n,m) represents electric field harmonics shown in FIG. 4B (405a˜d) and the asterisk denotes complex-conjugation operation.

As these harmonic terms are theoretically functions of the amplitude andphase of the incident electromagnetic field, a direct comparison betweenthe measured harmonics and the estimated harmonics can be made. Morespecifically, one can optimize point-by-point values of amplitude andphase by minimizing the error metric defined as

min_(φ,A)∫∫Σ_(m=±1,n=±1) |H _(m,n) ^(estimated) A,φ)−H _(m,n)^(measured)|² dxdy.  (6)

The starting value of the amplitude and phase maps are given by thefirst step reduction [Eqs. (3-4)]. The point-by-point optimizationproceeds from the low-resolution wavefront map obtained from the firststep to the estimation of a higher resolution map by directly matchingthe more exact mathematical expression of the given measured quantity.In practice, only non-diagonally paired harmonics are used because argH_(1,1)=−arg H_(−1,−1), and arg H_(−1,1)=−arg H_(1,−1) from the Fouriertransform theorem of a real function. Therefore the number of harmonicsto be handled is reduced from four to two in some embodiments by thissimplification. The point-by-point optimization process utilized hereinprovides a higher resolution phase map than the first-order zonal methodrepresented by Eq. (4) and other modal phase reconstruction methods.

FIG. 5A is a diagram illustrating a unit cell of the checkerboardpattern transmission mask illustrated in FIG. 4A. As illustrated in FIG.5A, an amplitude-only checkerboard transmission pattern mask can bemanufactured by etching a chrome layer on a 19.1 mm diameterfused-silica substrate. As will be evident to one of skill in the art,other low transmission layers on other substrates can be utilized asappropriate to the particular implementation. The dimension of the unitcell as shown in FIG. 5A is 60 μm square (i.e., Λ_(x)=Λ_(y)=Λ=60 μm),although other dimensions can be utilized, for example, up to about 10times larger than the pixel size of the detector used to detect theinterferogram. Additionally, unit cells utilizing rectangular (i.e.,Λ_(x) and Λ_(y)), circular, and other shapes are included within thescope of the present invention. In one embodiment, the patterned maskwas mounted directly in front of a 12-bit CCD camera.

FIG. 5B is a frequency domain plot associated with a measuredinterferogram associated with the rotated checkerboard patterntransmission mask illustrated in FIG. 4A/FIG. 5A. The interferogram 108obtained using the checkerboard pattern transmission mask was Fouriertransformed and the interferogram in the Fourier domain, also referredto as a frequency domain interferogram, is shown in FIG. 5B. As will benoted, the mask was rotated with respect to the CCD axes during themeasurement, resulting in the peaks in the measurement not aligning withthe x and y axes. Referring to FIGS. 4C and 5B, harmonic 407 acorresponds to H_(1,1), harmonic 407 b corresponds to H_(−1,1), harmonic407 c corresponds to H_(−1,−1), and harmonic 407 d corresponds toH_(1,−1).

In FIG. 5B, the spatial-frequency axes are divided by unit frequencyK_(0,x[y])=2π/Λ_(x[y]). The scale is compressed to bring out details ofsidelobe wing structures. Using embodiments of the present invention, itis unnecessary to carefully align the mask parallel to the CCD axessince the rotation angle can be calculated by, for example, the centroidof the H_(1,1). The slopes are then rotated into the CCD axes using theangle as described in relation to FIG. 8. During operation, spotsappearing at unexpected locations can be attributed to secondaryreflections in the system.

As described more fully below, the frequency of the harmonics, forexample, harmonic H_(1,1) will be a function of the carrier frequency ofthe mask and a small change in frequency resulting from non-uniformphase in the incident wavefront. For a wavefront with a uniform phasefront, each of the harmonics H_(1,1), H_(−1,1), H_(−1,−1), and H_(1,−1)would be represented by a frequency profile similar to a sync function,an Airy function, or the like depending on the spatial shape and finiteextent of the incident wavefront. As the wavefront varies from a uniformphase front, the frequency content of the harmonics will be blurred as aresult of the local modulation. The measurement and analysis of thisblurring is then used to determine the characteristics of the incidentwavefront.

The Fourier-domain interferogram in FIG. 5B shows faint dark linesbetween the sidelobes. Two dashed white lines are overlaid on theselines to aid in their identification. These lines are caused by thedestructive interference at the quarter-Talbot distance, referred toherein as Talbot lines. In some embodiments, it is advantageous to setthe distance L between the mask and the detector optically downstream ofthe mask so that the number of the Talbot lines are minimized and theirlocation is centered between the origin and the H_(±1,±1) peaks tomaximize spatial bandwidth. In the data shown in FIG. 5B, there are twoTalbot lines. The distance from the origin to the nearest Talbot linecan be computed as K_(L)=πΛ/(2)^(1/2)λL. L can be roughly estimatedusing K_(L) or can be more precisely determined by using a knownwavefront-calibration source.

The inventors have determined that positioning of the detector withrespect to the mask at a distance

${L = \frac{\Lambda^{2}}{8\lambda}},\frac{\Lambda^{2}}{4\lambda},\frac{3\Lambda^{2}}{4\lambda},{\frac{5\Lambda^{2}}{4\lambda}\mspace{14mu} \ldots}$

results in destructive interference between the first and zeroth orderdiffracted beams. As a result, the fringe visibility is very low atthese distances. Accordingly, in order to improve performance, Λ²/82 orthe odd-integer multiples of Λ²/4λ can be avoided by positioning themask and detector such that the distance between them is in accordancewith

${0 < L < {\frac{1}{8}\frac{\Lambda^{2}}{\lambda}\mspace{14mu} {or}\mspace{14mu} \frac{1}{4}\frac{\Lambda^{2}}{\lambda}( {{2n} - 1} )} < L < {\frac{1}{4}\frac{\Lambda^{2}}{\lambda}( {{2n} + 1} )}},$

where λ is the wavelength of the incident field and n is a positiveinteger (1, 2, 3, . . . ). In some embodiments, L is set at a distancethat is in a range of values between the distances at which destructiveinterference occurs. One of ordinary skill in the art would recognizemany variations, modifications, and alternatives.

As will be evident to one of skill in the art, L is not fixed for agiven wavelength, but can be defined by a range. A fixed L can beutilized for broad range of wavelengths with L satisfying the inequalityconditions, or in other words, L is a predetermined distance away fromthe distances associated with destructive interference.

As an example, for incident light in the wavelength range from 10 nm to2,000 nm, including soft-x-ray, ultraviolet, visible, and near-infraredspectra, L is typically in the range of 1 mm≦L≦10 mm. Although someembodiments are described in relation to optical wavelengths,embodiments of the present invention are not limited to opticalapplications and other wavelength regions, including microwave, areincluded within the scope of the present invention.

FIG. 6A is a plot illustrating a wavefront measurement made using anembodiment of the present invention. FIG. 6B is a plot illustrating awavefront measurement made using a conventional Shack-Hartmann sensor.FIG. 6C is a plot illustrating the difference between the plots in FIGS.6A and 6B. Referring to FIGS. 6A and 6B, a random wavefront wasgenerated by a spatial-light modulator and reconstructed wavefronts weremeasured using an embodiment of the present invention (FIG. 6A) andusing a Shack-Hartmann sensor (FIG. 6B). As illustrated in FIG. 6C, thedifference in the phase fronts between the two wavefront maps is 0.19waves in peak-to-valley and 0.03 waves in root mean square (rms) at1.053 μm wavelength, which demonstrates that embodiments of the presentinvention provide performance comparable to Shack-Hartmann sensors.

FIG. 7 is a diagram illustrating the use of an embodiment of the presentinvention in aligning hexagonally segmented mirrors according to anembodiment of the present invention. Hexagonally segmented mirrors areoften used in large astronomical telescopes. Three regions of thecheckerboard transmission mask are printed on a single substrate. Theorientation and the location of these regions are designed such thatthey overlap the boundaries or seams of the hexagonal mirror segments inthe image of the hexagonal mirrors. Referring to FIG. 7, a mask 701could include (three) amplitude-only checkerboard patterns 102 a, 102 b,and 102 c oriented in different directions on a single plate. The dottedhexagons 705 represent segmented hexagonal spherical mirrors. Ingeneral, the wavefront reflected from these segmented surfaces is notcontinuous due to piston differences between the hexagonal segments. Theinterference from each pattern could be used to co-phase the threemirrors. One of ordinary skill in the art would recognize manyvariations, modifications, and alternatives.

FIG. 8 is a method of measuring a wavefront according to an embodimentof the present invention. As described below, the method can be used tocompute a phase front profile of an incident beam. The method 800includes obtaining an interferogram associated with an incident beam,characterized by a wavefront, passing through a transmission mask (810).The transmission mask can be a checkerboard pattern transmission mask,for example, an amplitude-only transmission mask. The interferogram canbe represented by the intensity proportional to the value computed inEq. (2).

An exemplary interferogram is illustrated as interferogram 108 in FIGS.1 and 2. The method also includes Fourier transforming the interferogramto provide a frequency domain interferogram (812). Referring to FIG. 5B,the frequency domain plot associated with a measured interferogramassociated with the Checkerboard pattern transmission mask is anexemplary frequency domain interferogram.

As illustrated in FIG. 5B, the frequency domain interferogram includes anumber of harmonics. The method includes selecting a subset of harmonicsfrom the frequency domain interferogram (814) and individually inverseFourier transforming each of the subset of harmonics to provide a set ofspatial domain harmonics (816). During the selection process, the subsetof harmonics can be cropped to remove contributions from higher orderharmonics and to reduce the number of samples for faster processing.Referring to FIG. 5B, the harmonics H_(1,1), H_(−1,1), H_(−1,−1), andH_(1,−1) are circled to illustrate cropping of the harmonics in thefrequency domain. In an embodiment, the subset of harmonics includes theharmonics closest to the origin in each quadrant (e.g., H_(1,1),H_(−1,1), H_(−1,1), and H_(1,−1)). In another embodiment, the subset ofharmonics consists of the harmonics closest to the origin in twoadjacent quadrants (either H_(1,1), and H_(−1,1) or H_(−1,1) andH_(1,−1)). In some embodiments, the frequency domain interferogram isrepresented by a matrix and smaller matrices centered on the harmonicsare utilized to perform the cropping function. One of ordinary skill inthe art would recognize many variations, modifications, andalternatives.

After selection and cropping of the harmonics, a down-sampled set ofharmonics is available for use in performing the inverse Fouriertransform. As will be evident to one of skill in the art, the centeringof the harmonics during the cropping process removes the carrierfrequency, resulting in spatial domain down-sampled harmonics, referredto as spatial domain harmonics, as a result of the inverse Fouriertransform. The result can be considered as an array of complex valuesrelated to the amplitude and phase of the incident wavefront.

The method further includes extracting a phase profile from each of theset of spatial domain harmonics (818). For each harmonic, the phaseprofile will represent a linear combination of the slopes of the phasein the x and y-directions as discussed in relation to Eq. (4), moreparticularly, the inversion of Eq. (4). FIG. 10A is a plot illustratinga wrapped phase profile for a first harmonic according to an embodimentof the present invention. FIG. 10B is a plot illustrating an unwrappedphase profile for the first harmonic according to an embodiment of thepresent invention. FIG. 10C is a plot illustrating a rotated phaseprofile for the first harmonic according to an embodiment of the presentinvention.

As illustrated in FIG. 10A, the slope of the phase of harmonic H_(1,1)is illustrated in the x-direction and the y-direction. For reference,refer to harmonic 407 a in FIG. 4C and harmonic H_(1,1) in FIG. 5B. Thediscontinuity (i.e., 2π shift) in the slope is illustrated by thetransition between black and white in the top left half of the plot. Themethod includes removing phase discontinuities in the phase profile(820) and rotating the phase profile (822). Referring to FIG. 10B, thephase discontinuity in the top left portion of the plot has been removedduring the unwrapping process so that the phase profile is continuousover the plot. By removing these phase discontinuities, embodiments ofthe present invention enable embodiments of the present invention tomeasure phase fronts characterized by large variation in phase acrossthe wavefront, which contrasts with approaches that do not use the phaseunwrapping process, which are unable to measure large variation phasefronts and are limited to small variations in the phase front.

FIG. 10C illustrate the special case of quadratic input wavefront. Thephase profile, i.e., the slope of the input wavefront after rotation isin a single direction. As illustrated in FIG. 10C, the phase varies inthe x-direction, but is constant in the y-direction for H_(1,1).Similarly, in FIG. 11C, the phase varies in the y-direction, but isconstant in the x-direction for H_(−1,1). The rotation of the phaseprofile simplifies system alignment since rotation of the transmissionmask with respect to the detector can be removed by the system.

FIG. 11A is a plot illustrating a phase profile for a second harmonicaccording to an embodiment of the present invention. For reference,refer to harmonic 407 b in FIG. 4C and harmonic H_(−1,1) in FIG. 5B.FIG. 11B is a plot illustrating an unwrapped phase profile for thesecond harmonic according to an embodiment of the present invention.FIG. 11C is a plot illustrating a rotated phase profile for the secondharmonic according to an embodiment of the present invention. In FIGS.11A-11C, the harmonic used is the H_(−1,1) harmonic. The phasediscontinuity in the top right portion of FIG. 11A is removed byunwrapping the phase profile to provide the plot in FIG. 11B. Therotation of the phase profile results in the phase profile in FIG. 11Cthat only varies in the y-direction. It will be noted that the ramps inthe phase profiles for H_(1,1) and H_(−1,1) are orthogonal to eachother.

The quantities represented by FIGS. 10B and 11B are the wavefront slopesviewed from the checkerboard mask frame of reference and the quantitiesrepresented by FIGS. 10C and 11C are viewed from the detector frame ofreference. The quantities represented by FIGS. 10C and 11C are theresults of the linear transformation of the quantities represented byFIGS. 10B and 11B.

Using phase profiles of harmonics, the x and y derivatives of theincident beam phase front profile are calculated. From thesederivatives, the phase front profile is reconstructed by integrating therotated phase profiles (822). In embodiments, reconstructing the phasefront profile includes performing a two-dimensional integration of thephase profiles associated with each of the set of spatial domainharmonics.

The method includes, in some embodiments, computing the intensityprofile associated with the incident beam. In these embodiments, themethod includes measuring an amplitude associated with the spatialdomain harmonics and computing an intensity profile associated withincident beam. The amplitude associated with the spatial domainharmonics is computed based on the absolute value of the array ofcomplex values and the intensity is a function of the square of theamplitude values for the harmonics as illustrated in Eq. (3). Byproviding the intensity profile of the incident wavefront, embodimentsof the present invention provide data not typically available usingconventional wavefront sensors, which only provide the low-resolutionintensity profile and need an extra camera to provide high-resolutionintensity data.

It should be appreciated that the specific steps illustrated in FIG. 8provide a particular method of measuring a wavefront according to anembodiment of the present invention. Other sequences of steps may alsobe performed according to alternative embodiments. For example,alternative embodiments of the present invention may perform the stepsoutlined above in a different order. Moreover, the individual stepsillustrated in FIG. 8 may include multiple sub-steps that may beperformed in various sequences as appropriate to the individual step.Furthermore, additional steps may be added or removed depending on theparticular applications. One of ordinary skill in the art wouldrecognize many variations, modifications, and alternatives.

FIG. 9 is a schematic diagram illustrating a wavefront sensor accordingto an embodiment of the present invention. The wavefront sensor includesa transmission mask 910 upon which an incident beam is incident. Theincident beam is characterized by a wavefront defined by an intensityprofile and a phase front profile. The incident beam is diffracted as itpasses through the transmission mask 910 and impinges on an imagingdevice 912, such as a CCD camera. The interferogram produced by thelight diffracting through the transmission mask is measured using theimaging device. The computer 950 receives the interferogram and computesthe wavefront of the incident beam, including the phase front profileand/or the intensity profile.

The computer 950 includes a processor 952, also referred to as a dataprocessor, a storage device 954, and an input/output device 956. Theprocessor 952 represents a central processing unit of any type ofarchitecture, such as a CISC (Complex Instruction Set Computing), RISC(Reduced Instruction Set Computing), VLIW (Very Long Instruction Word),or a hybrid architecture, although any appropriate processor may beused. The processor 952 executes instructions and includes that portionof the computer 950 that controls the operation of the entire computer.Although not depicted in FIG. 9, the processor 952 typically includes acontrol unit that organizes data and program storage in memory andtransfers data and other information between the various parts of thecomputer 950. The processor 952 receives input data from theinput/output module 956 and reads and stores code and data in thestorage device 954 and presents data to the input/output module 956 andthe user interface 958.

Although the computer 950 is shown to contain only a single processor952, the disclosed embodiment applies equally to computers that may havemultiple processors and to computers that may have multiple buses withsome or all performing different functions in different ways.

The storage device 954 represents one or more mechanisms for storingdata. For example, the storage device 954 may include cloud storage,read-only memory (ROM), random access memory (RAM), magnetic diskstorage media, optical storage media, flash memory devices, and/or othermachine-readable media. In other embodiments, any appropriate type ofstorage device may be used. Although only one storage device 954 isshown, multiple storage devices and multiple types of storage devicesmay be present. Further, although the computer 950 is drawn to containthe storage device 954, it may be distributed across other computers,for example on a server or otherwise in the cloud.

The storage device 954 includes a controller (not shown in FIG. 9) anddata items. The controller includes instructions capable of beingexecuted on the processor 952 to carry out the methods described morefully throughout the present specification, including Fouriertransforms, inverse Fourier transforms, integration of the phaseprofiles, and the like. In another embodiment, some or all of thefunctions are carried out via hardware in lieu of a processor-basedsystem. In one embodiment, the controller is a web browser, but in otherembodiments the controller may be a database system, a file system, anelectronic mail system, a media manager, an image manager, or mayinclude any other functions capable of accessing data items. Of course,the storage device 954 may also contain additional software and data(not shown), which is not necessary to understand the invention.

The embodiments described herein may be implemented in an operatingenvironment comprising software installed on any programmable device, inhardware, or in a combination of software and hardware. Althoughembodiments have been described with reference to specific exampleembodiments, it will be evident that various modifications and changesmay be made to these embodiments without departing from the broaderspirit and scope of the invention. Accordingly, the specification anddrawings are to be regarded in an illustrative rather than a restrictivesense.

As may be used herein for purposes of the present disclosure, the term‘about’ means the amount of the specified quantity plus/minus afractional amount thereof that a person skilled in the art wouldrecognize as typical and reasonable for that particular quantity ormeasurement; e.g., “wherein Λ and Λ_(y) are about 10 times larger than apixel size of a detector used to detect the interferogram” could mean‘wherein Λ_(x)=Λ_(y) is equal to about 60±8 μm for a 6.45 μm pixel CCDchip.’ Likewise, the term ‘substantially’ means as close to or similarto the specified term being modified as a person skilled in the artwould recognize as typical and reasonable; for e.g., within typicalmanufacturing and/or assembly tolerances, as opposed to beingintentionally different by design and implementation.

It should be appreciated that all combinations of the foregoing conceptsand additional concepts discussed herein (provided such concepts are notmutually inconsistent) are contemplated as being part of the inventivesubject matter disclosed herein. In particular, all combinations ofclaimed subject matter appearing at the end of this disclosure arecontemplated as being part of the inventive subject matter disclosedherein. It should also be appreciated that terminology explicitlyemployed herein that also may appear in any disclosure incorporated byreference should be accorded a meaning most consistent with theparticular concepts disclosed herein.

It is also understood that the examples and embodiments described hereinare for illustrative purposes only and that various modifications orchanges in light thereof will be suggested to persons skilled in the artand are to be included within the spirit and purview of this applicationand scope of the appended claims.

What is claimed is:
 1. A method of measuring characteristics of awavefront of an incident beam, the method comprising: obtaining aninterferogram associated with the incident beam passing through atransmission mask; Fourier transforming the interferogram to provide afrequency domain interferogram; selecting a subset of harmonics from thefrequency domain interferogram; individually inverse Fouriertransforming each of the subset of harmonics to provide a set of spatialdomain harmonics; extracting a phase profile from each of the set ofspatial domain harmonics; removing phase discontinuities in the phaseprofile; rotating the phase profile; and reconstructing a phase front ofthe wavefront of the incident beam.
 2. The method of claim 1 furthercomprising: measuring an amplitude associated with the spatial domainharmonics; and computing an intensity profile associated with thewavefront of the incident beam.
 3. The method of claim 1 wherein thetransmission mask comprises a checkerboard pattern transmission mask. 4.The method of claim 3 wherein the checkerboard pattern transmission maskcomprises an amplitude-only transmission mask.
 5. The method of claim 1wherein the subset of harmonics comprises a harmonic closest to anorigin in each quadrant.
 6. The method of claim 1 wherein the subset ofharmonics comprises a harmonic closest to the origin in two adjacentquadrants.
 7. The method of claim 1 wherein the phase profilecharacterizes a slope of the phase front.
 8. The method of claim 1wherein reconstructing the phase front comprises performing an twodimensional integration of the phase profiles associated with each ofthe set of spatial domain harmonics.
 9. A method for performing opticalwavefront sensing, the method comprising: providing an amplitudetransmission mask having a light input side, a light output side, and anoptical transmission axis passing from the light input side to the lightoutput side, wherein the amplitude transmission mask is characterized bya checkerboard pattern having a square unit cell of size Λ, directing anincident light field having a wavelength λ to be incident on the lightinput side; propagating the incident light field through the amplitudetransmission mask; producing a plurality of diffracted light fields onthe light output side; and detecting, at a detector disposed a distanceL from the amplitude transmission mask, an interferogram associated withthe plurality of diffracted light fields; wherein:${0 < L < {\frac{1}{8}\frac{\Lambda^{2}}{\lambda}\mspace{14mu} {or}\mspace{14mu} \frac{1}{4}\frac{\Lambda^{2}}{\lambda}( {{2n} - 1} )} < L < {\frac{1}{4}\frac{\Lambda^{2}}{\lambda}( {{2n} + 1} )}},$where n is an integer greater than zero.
 10. The method of claim 9wherein the amplitude transmission mask comprises an amplitude-onlytransmission mask.
 11. The method of claim 9 wherein the detector ischaracterized by a pixel size and A is about 10 times larger than thepixel size.
 12. The method of claim 9 wherein:10 nm≦λ≦2,000 nm; and1 mm≦L≦10 mm.
 13. The method of claim 9 wherein the interferogramincludes phase slope information, the method further comprisingintegrating a two-dimensional phase slope computed using theinterferogram to construct a phase front associated with the incidentlight field.
 14. The method of claim 13 wherein constructing the phasefront comprises performing a Fourier domain spatial carrier analysis.15. The method of claim 14 wherein the Fourier domain spatial carrieranalysis comprises selecting carrier terms located diagonally, but notorthogonally, with respect to axes of the amplitude transmission mask.16. The method of claim 15 further comprising performing apoint-by-point optimization on the diagonal carrier terms to refine aspatial resolution of the wavefront.
 17. The method of claim 9 furthercomprising: providing a segmented mirror array; and providing aplurality of the amplitude-only transmission masks adjacent thesegmented mirror array such that each one of the amplitude-onlytransmission masks is disposed along a seam of two adjoining mirrorsegments.
 18. The method of claim 17 wherein the mirror segments arehexagonal.
 19. A wavefront sensor comprising: an amplitude-onlytransmission mask characterized by a checkerboard pattern having a unitsquare cell of size Λ; a detector disposed at a distance, L, opticallydownstream of the amplitude-only transmission mask, wherein${0 < L < {\frac{1}{8}\frac{\Lambda^{2}}{\lambda}\mspace{14mu} {or}\mspace{14mu} \frac{1}{4}\frac{\Lambda^{2}}{\lambda}( {{2n} - 1} )} < L < {\frac{1}{4}\frac{\Lambda^{2}}{\lambda}( {{2n} + 1} )}},$where λ is the wavelength of the incident field and n is a positiveinteger; and a computer coupled to the detector.
 20. The wavefrontsensor of claim 19 wherein the computer includes a non-transitorycomputer-readable storage medium comprising a plurality ofcomputer-readable instructions tangibly embodied on thecomputer-readable storage medium, which, when executed by a dataprocessor, provide a measurement of characteristics of a wavefront of anincident beam, the plurality of instructions comprising: instructionsthat cause the data processor to obtain an interferogram associated withthe incident beam passing through a transmission mask; instructions thatcause the data processor to Fourier transform the interferogram toprovide a frequency domain interferogram; instructions that cause thedata processor to select a subset of harmonics from the frequency domaininterferogram; instructions that cause the data processor toindividually inverse Fourier transform each of the subset of harmonicsto provide a set of spatial domain harmonics; instructions that causethe data processor to extract a phase profile from each of the set ofspatial domain harmonics; instructions that cause the data processor toremove phase discontinuities in the phase profile; instructions thatcause the data processor to rotate the phase profile; and instructionsthat cause the data processor to reconstruct a phase front of thewavefront of the incident beam.